Hysteresis definition is as follow:
the phenomenon in which the value of a physical property lags behind changes in the effect causing it, as for instance when magnetic induction lags behind the magnetizing force.
In asymmetric bipolar power supply (such as those created by elevating a virtual ground from unipolar power source), the hysteresis can be skewed along the horizontal axis of the above transfer function.
Schmitt triggers vs. Closed-loop amplifiers
It is easy to convert a closed-loop amplifier into Schmitt trigger by swapping the input terminals. Standard closed-loop amplifiers have negative feedbacks while Schmitt triggers have positive feedbacks as defined by Wikipedia:
In electronics, a Schmitt trigger is a comparator circuit with hysteresis implemented by applying positive feedback to the noninverting input of a comparator or differential amplifier.
One important note when converting negative feedback comparators into Schmitt triggers is the direction of the output. As seen below, non-inverting amplifiers will yield inverting Schmitt triggers when the input terminals are swapped, and vice versa.
Fortunately, changing the output direction of a comparator circuit (any feedback) is quite simple. Swapping Vin and GND terminals inverses the output direction of the circuit.
Schmitt trigger’s characteristics
Schmitt triggers are commonly used for switch debouncing and noise filtering for digital signals. In digital circuits, noisy signals are often fed to a low-pass filter to create a smoother signal before passing through a Schmitt trigger to recreate the sharp digital signal. Despite its importance in digital circuitry, the trigger itself is an analog component and is sometimes omitted in digital circuit simulation libraries.
Due to open-loop nature of the circuit, the gain of Schmitt trigger is infinity. Impedance values follow normal op-amp characteristics.
The regenerative feedback refers to the portion coming out of the voltage divider and into the non-inverting input. It’s denoted with beta symbol and defined as follow:
B = R1 / (R1 + R2)
The switching thresholds are calculated as the regenerative feedback times the positive or negative supply rail voltage Vs as seen below
Please note that despite the formula, in reality, the Vs value in this calculation experiences some voltage drops (around 15%) due to internal impedance; hence the switching thresholds might be lower than calculated.
Astable multivibrator (Relaxation oscillator)
With a few extra components (a capacitor and a resistor), a Schmitt trigger can be adapted into an astable multivibrator. From inverting Schmitt trigger, add an additional negative feedback using a resistor, a capacitor, and ground to create the astable multivibrator.
The multivibrator does not take in an input signal. When powered up, it creates a full-range square waveform across the op-amp’s output and a half-range waveform across the non-inverting input (some documents denote this waveform as the reference voltage). At the same time, it creates a half-range ramp waveform across the capacitor.
The time period of the multivibrator is given by
T = 2 RC ln[(1+B)/(1-B)]
with B is the regenerative feedback, R and C are the value of the resistor and the value of the capacitor across the negative feedback respectively.
The amplitudes of half-range voltages are calculated using the same formula as the switching thresholds of normal Schmitt triggers.
When using an asymmetric bipolar supply, the switching thresholds of the multivibrator will be skewed and it will produce a waveform of adjustable duty cycles. The duty cycle is dependent on the offset; left offset produces shorter duty cycle and right offset produces longer duty cycle. However, the device can only accept so much power supply offsetting before it cannot operate (under load) or overloaded.